Since a measuring instrument cannot be rigidly fixed to the spatial reference frame and, at the same time, be movable relative to it, the experiments which serve to precisely determine the position and the momentum of an object are mutually exclusive. Of course, in itself, this is not at all typical for quantum mechanics.
But, because the interaction between object and instrument during the measurement can neither be neglected nor determined the two measurements cannot be combined. This means that in the description of the object one must choose between the assignment of a precise position or of a precise momentum. Similar considerations hold with respect to the measurement of time and energy. Just as the spatial coordinate system must be fixed by means of solid bodies so must the time coordinate be fixed by means of unperturbed, synchronised clocks.
But it is precisely this requirement which prevents one from taking into account of the exchange of energy with the instrument if this is to serve its purpose. Conversely, any conclusion about the object based on the conservation of energy prevents following its development in time. The conclusion is that in quantum mechanics we are confronted with a complementarity between two descriptions which are united in the classical mode of description: the space-time description or coordination of a process and the description based on the applicability of the dynamical conservation laws.
Conversely, any attempt of locating the collision between the photon and the electron more accurately would, on account of the unavoidable interaction with the fixed scales and clocks defining the space-time reference frame, exclude all closer account as regards the balance of momentum and energy. A causal description of the process cannot be attained; we have to content ourselves with complementary descriptions.
In addition to complementary descriptions Bohr also talks about complementary phenomena and complementary quantities. Position and momentum, as well as time and energy, are complementary quantities. By comparison, he seemed to put little value on arguments starting from the mathematical formalism of quantum theory.
One might say that for Bohr the conceptual clarification of the situation has primary importance while the formalism is only a symbolic representation of this situation.
This is remarkable since, finally, it is the formalism which needs to be interpreted. We close this section by citing from an article of to show how Bohr conceived the role of the formalism of quantum mechanics:. The entire formalism is to be considered as a tool for deriving predictions, of definite or statistical character, as regards information obtainable under experimental conditions described in classical terms and specified by means of parameters entering into the algebraic or differential equations of which the matrices or the wave-functions, respectively, are solutions.
These symbols themselves, as is indicated already by the use of imaginary numbers, are not susceptible to pictorial interpretation; and even derived real functions like densities and currents are only to be regarded as expressing the probabilities for the occurrence of individual events observable under well-defined experimental conditions. In his Como lecture, published in , Bohr gave his own version of a derivation of the uncertainty relations between position and momentum and between time and energy.
He started from the relations. He noticed that a wave packet of limited extension in space and time can only be built up by the superposition of a number of elementary waves with a large range of wave numbers and frequencies.
The original text has equality signs instead of approximate equality signs, but, since Bohr does not define the spreads exactly the use of approximate equality signs seems more in line with his intentions.
Moreover, Bohr himself used approximate equality signs in later presentations. These equations determine, according to Bohr:. This circumstance may be regarded as a simple symbolic expression of the complementary nature of the space-time description and the claims of causality.
First of all, Bohr does not refer to discontinuous changes in the relevant quantities during the measurement process. Rather, he emphasizes the possibility of defining these quantities. A draft version of the Como lecture is even more explicit on the difference between Bohr and Heisenberg:. These reciprocal uncertainty relations were given in a recent paper of Heisenberg as the expression of the statistical element which, due to the feature of discontinuity implied in the quantum postulate, characterizes any interpretation of observations by means of classical concepts.
It must be remembered, however, that the uncertainty in question is not simply a consequence of a discontinuous change of energy and momentum say during an interaction between radiation and material particles employed in measuring the space-time coordinates of the individuals.
According to the above considerations the question is rather that of the impossibility of defining rigorously such a change when the space-time coordination of the individuals is also considered. The unaccustomed features of the situation with which we are confronted in quantum theory necessitate the greatest caution as regard all questions of terminology. Speaking, as it is often done of disturbing a phenomenon by observation, or even of creating physical attributes to objects by measuring processes is liable to be confusing, since all such sentences imply a departure from conventions of basic language which even though it can be practical for the sake of brevity, can never be unambiguous.
Nor did he approve of an epistemological formulation or one in terms of experimental inaccuracies:. Bohr ; also Bohr According to such a formulation it would appear as though we had to do with some arbitrary renunciation of the measurement of either the one or the other of two well-defined attributes of the object, which would not preclude the possibility of a future theory taking both attributes into account on the lines of the classical physics. Instead, Bohr always stressed that the uncertainty relations are first and foremost an expression of complementarity.
This may seem odd since complementarity is a dichotomic relation between two types of description whereas the uncertainty relations allow for intermediate situations between two extremes. But they also allow intermediate situations in which the mentioned uncertainties are all non-zero and finite.
This more positive aspect of the uncertainty relation is mentioned in the Como lecture:. At the same time, however, the general character of this relation makes it possible to a certain extent to reconcile the conservation laws with the space-time coordination of observations, the idea of a coincidence of well-defined events in space-time points being replaced by that of unsharply defined individuals within space-time regions.
However, Bohr never followed up on this suggestion that we might be able to strike a compromise between the two mutually exclusive modes of description in terms of unsharply defined quantities. Indeed, an attempt to do so, would take the formalism of quantum theory more seriously than the concepts of classical language, and this step Bohr refused to take.
Instead, in his later writings he would be content with stating that the uncertainty relations simply defy an unambiguous interpretation in classical terms:. These so-called indeterminacy relations explicitly bear out the limitation of causal analysis, but it is important to recognize that no unambiguous interpretation of such a relation can be given in words suited to describe a situation in which physical attributes are objectified in a classical way.
These two approaches are equivalent as far as the relationship between position and momentum is concerned, but this is not so for time and energy since most physical systems do not have a time operator. Indeed, in his discussion with Einstein Bohr , Bohr considered time as a simple classical variable.
For more details see Hilgevoord and In the previous two sections we have seen how both Heisenberg and Bohr attributed a far-reaching status to the uncertainty relations. They both argued that these relations place fundamental limits on the applicability of the usual classical concepts.
Moreover, they both believed that these limitations were inevitable and forced upon us. However, we have also seen that they reached such conclusions by starting from radical and controversial assumptions. This entails, of course, that their radical conclusions remain unconvincing for those who reject these assumptions.
Indeed, the operationalist-positivist viewpoint adopted by these authors has long since lost its appeal among philosophers of physics. So the question may be asked what alternative views of the uncertainty relations are still viable. Of course, this problem is intimately connected with that of the interpretation of the wave function, and hence of quantum mechanics as a whole.
Since there is no consensus about the latter, one cannot expect consensus about the interpretation of the uncertainty relations either. In quantum mechanics a system is supposed to be described by its wave function, also called its quantum state or state vector. The operational meaning of these probability distributions is that they correspond to the distribution of the values obtained for these quantities in a long series of repetitions of the measurement. More precisely, one imagines a great number of copies of the system under consideration, all prepared in the same way.
On each copy the momentum, say, is measured. Generally, the outcomes of these measurements differ and a distribution of outcomes is obtained. The theoretical momentum distribution derived from the quantum state is supposed to coincide with the hypothetical distribution of outcomes obtained in an infinite series of repetitions of the momentum measurement.
The same holds, mutatis mutandis , for all the other physical quantities pertaining to the system. Note that no simultaneous measurements of two or more quantities are required in defining the operational meaning of the probability distributions. The uncertainty relations discussed above can be considered as statements about the spreads of the probability distributions of the several physical quantities arising from the same state.
Inequality 9 is an example of such a relation in which the standard deviation is employed as a measure of spread. From this characterization of uncertainty relations follows that a more detailed interpretation of the quantum state than the one given in the previous paragraph is not required to study uncertainty relations as such. In particular, a further ontological or linguistic interpretation of the notion of uncertainty, as limits on the applicability of our concepts given by Heisenberg or Bohr, need not be supposed.
Of course, this minimal interpretation leaves the question open whether it makes sense to attribute precise values of position and momentum to an individual system. Some interpretations of quantum mechanics, e. The only requirement is that, as an empirical fact, it is not possible to prepare pure ensembles in which all systems have the same values for these quantities, or ensembles in which the spreads are smaller than allowed by quantum theory.
Although interpretations of quantum mechanics, in which each system has a definite value for its position and momentum are still viable, this is not to say that they are without strange features of their own; they do not imply a return to classical physics. We end with a few remarks on this minimal interpretation. First, it may be noted that the minimal interpretation of the uncertainty relations is little more than filling in the empirical meaning of inequality 9.
As such, this view shares many of the limitations we have noted above about this inequality. Indeed, it is not straightforward to relate the spread in a statistical distribution of measurement results with the inaccuracy of this measurement, such as, e. Moreover, the minimal interpretation does not address the question whether one can make simultaneous accurate measurements of position and momentum. As a matter of fact, one can show that the standard formalism of quantum mechanics does not allow such simultaneous measurements.
But this is not a consequence of relation 9. Rather, it follows from the fact that this formalism simply does not contain any observable that would accomplish such a task. If one feels that statements about inaccuracy of measurement, or the possibility of simultaneous measurements, belong to any satisfactory formulation of the uncertainty principle, one will need to look for other formulations of the uncertainty principle.
Some candidates for such formulations will be discussed in Section 6. First, however, we will look at formulations of the uncertainty principle that stay firmly within the minimal interpretation, and differ from 9 only by using measures of uncertainty other than the standard deviation. While the standard deviation is the most well-known quantitative measure for uncertainty or the spread in the probability distribution, it is not the only one, and indeed it has distinctive drawbacks that other such measures may lack.
This means, in our view, that relation 9 actually fails to express what most physicists would take to be the very core idea of the uncertainty principle. One way to deal with this objection is to consider alternative measures to quantify the spread or uncertainty associated with a probability density. Here we discuss two such proposals. Landau and Pollak obtained an uncertainty relation in terms of these bulk widths. Note that bulk widths are not so sensitive to the behavior of the tails of the distributions and, therefore, the Landau-Pollak inequality is immune to the objection above.
Thus, this inequality expresses constraints on quantum mechanical states not contained in relation 9. This, obviously, is not the best lower bound for the product of standard deviations, but the important point is here that the Landau-Pollak inequality 16 in terms of bulk widths implies the existence of a lower bound on the product of standard deviations, while conversely, the Heisenberg-Kennard equality 9 does not imply any bound on the product of bulk widths. A generalization of this approach to non-commuting observables in a finite-dimensional Hilbert space is discussed in Uffink Another approach to express the uncertainty principle is to use entropic measures of uncertainty.
A nice feature of this entropic uncertainty relation is that it provides a strict improvement of the Heisenberg-Kennard relation. A drawback of this relation is that it does not completely evade the objection mentioned above, i. Then we obtain the uncertainty relation Maassen and Uffink :. Both the standard deviation and the alternative measures of uncertainty considered in the previous subsection and many more that we have not mentioned!
Applied to quantum mechanics, where the probability distributions for position and momentum are obtained from a given quantum state vector, one can use them to formulate uncertainty relations that characterize the spread in those distribution for any given state. The resulting inequalities then express limitations on what state-preparations quantum mechanics allows.
They are thus expressions of what may be called a preparation uncertainty principle :. The relations 9 , 16 , 19 all belong to this category; the mere difference being that they employ different measures of spread: viz.
To create a measurement, an interaction with the particle must occur that will alter it's other variables. For example, in order to measure the position of an electron there must be a collision between the electron and another particle such as a photon. This will impart some of the second particle's momentum onto the electron being measured and thereby altering it. A more accurate measurement of the electron's position would require a particle with a smaller wavelength, and therefore be more energetic, but then this would alter the momentum even more during collision.
An experiment designed to determine momentum would have a similar effect on position. Consequently, experiments can only gather information about a single variable at a time with any amount of accuracy.
The mass of the water is 2 orders of magnitude smaller than that of the football, and the resulting position uncertainty is 2 orders of magnitude larger.
One example that can be used is a glass of water in a cup holder inside a moving car. This glass of water has multiple water molecules each consisting of electrons. The water in the glass is a macroscopic object and can be viewed with the naked eye.
The electrons however occupy the same space as the water, but cannot be seen and therefore must be measured microscopically. As stated above in the introduction, the effect of measuring a tiny particle causes a change in its momentum and time in space, but this is not the case for the larger object. Thus, the uncertainty principle has much greater bearing on the electrons rather than the macroscopic water.
Introduction Heisenberg's Uncertainty Principle states that there is inherent uncertainty in the act of measuring a variable of a particle. What does it mean? When and how does the model of many microscopic possibilities resolve itself into a particular macroscopic state? When and how does the fog bank of microscopic possibilities transform itself to the blurred picture we have of a definite macroscopic state. That is the collapse of the wave function problem and Schrodinger's cat is a simple and elegant explanation of that problem.
The macroscopic world is Newtonian and deterministic for local events note however that even the macroscopic world suffers from chaos. On the other hand, the microscopic quantum world radical indeterminacy limits any certainty surrounding the unfolding of physical events. Many things in the Newtonian world are unpredictable since we can never obtain all the factors effecting a physical system. But, quantum theory is much more unsettling in that events often happen without cause e.
Note that the indeterminacy of the microscopic world has little effect on macroscopic objects. This is due to the fact that wave function for large objects is extremely small compared to the size of the macroscopic world.
Your personal wave function is much smaller than any currently measurable sizes. And the indeterminacy of the quantum world is not complete because it is possible to assign probabilities to the wave function.
But, as Schrodinger's Cat paradox show us, the probability rules of the microscopic world can leak into the macroscopic world. The paradox of Schrodinger's cat has provoked a great deal of debate among theoretical physicists and philosophers.
Although some thinkers have argued that the cat actually does exist in two superposed states, most contend that superposition only occurs when a quantum system is isolated from the rest of its environment.
Various explanations have been advanced to account for this paradox--including the idea that the cat, or simply the animal's physical environment such as the photons in the box , can act as an observer. The question is, at what point, or scale, do the probabilistic rules of the quantum realm give way to the deterministic laws that govern the macroscopic world? This question has been brought into vivid relief by the recent work where an NIST group confined a charged beryllium atom in a tiny electromagnetic cage and then cooled it with a laser to its lowest energy state.
In this state the position of the atom and its "spin" a quantum property that is only metaphorically analogous to spin in the ordinary sense could be ascertained to within a very high degree of accuracy, limited by Heisenberg's uncertainty principle.
The workers then stimulated the atom with a laser just enough to change its wave function; according to the new wave function of the atom, it now had a 50 percent probability of being in a "spin-up" state in its initial position and an equal probability of being in a "spin-down" state in a position as much as 80 nanometers away, a vast distance indeed for the atomic realm.
In effect, the atom was in two different places, as well as two different spin states, at the same time--an atomic analog of a cat both living and dead. The clinching evidence that the NIST researchers had achieved their goal came from their observation of an interference pattern; that phenomenon is a telltale sign that a single beryllium atom produced two distinct wave functions that interfered with each other.
The modern view of quantum mechanics states that Schrodinger's cat, or any macroscopic object, does not exist as superpositions of existence due to decoherence. A pristine wave function is coherent, i. But Schrodinger's cat is not a pristine wave function, its is constantly interacting with other objects, such as air molecules in the box, or the box itself.
Thus a macroscopic object becomes decoherent by many atomic interactions with its surrounding environment. Decoherence explains why we do not routinely see quantum superpositions in the world around us. It is not because quantum mechanics intrinsically stops working for objects larger than some magic size. Instead, macroscopic objects such as cats and cards are almost impossible to keep isolated to the extent needed to prevent decoherence. Microscopic objects, in contrast, are more easily isolated from their surroundings so that they retain their quantum secrets and quantum behavior.
One of the surprising results of quantum physics is that if a physical event is not specifically forbidden by a quantum rule, than it can and will happen. While this may strange, it is a direct result of the uncertainty principle.
Things that are strict laws in the macroscopic world, such as the conversation of mass and energy, can be broken in the quantum world with the caveat that they can only broken for very small intervals of time less than a Planck time.
The violation of conservation laws led to the one of the greatest breakthroughs of the early 20th century, the understanding of radioactivity decay fission and the source of the power in stars fusion. More than 80 years after the uncertainty principle was first proposed, scientists are ironing out some uncertainties about the famous physics notion.
The uncertainty principle , proposed in by German physicist Werner Heisenberg, states that the more precisely a particle's position is measured, the less precisely its momentum can be known, and vice versa. It has long been invoked to describe the way measuring an object disturbs that object. Rozema and his colleagues found this aspect of the uncertainty principle is often misunderstood, and that quantum measurements don't wreak as much havoc on what they're measuring as many people, including physicists, assume.
0コメント